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On the stability of solutions to non-Newtonian Navier--Stokes--Fourier-like systems in the supercritical case

Anna Abbatiello, Miroslav Bulíček, Petr Kaplický

TL;DR

This work analyzes a three‑dimensional, incompressible non‑Newtonian, heat‑conducting fluid in a Lipschitz domain, modeled by a Navier–Stokes–Fourier–type system with nonuniform boundary temperature and a general $p$‑growth constitutive law $\mathbf{S}=\mathbf{S}^*(\vartheta,\mathbf{D}\mathbf{v})$ in the supercritical range. To handle potential lack of regularity and energy equality, the authors introduce a novel $b$‑weak solution framework built around a $\gamma$‑correction function and a $b$‑corrected total energy inequality, together with an entropy inequality, and prove global existence for $p\in(6/5,11/5)$ and $\gamma\in(-1,-1/2)$. They establish nonlinear stability of the unique steady state $(\mathbf{0},\hat{\vartheta})$ by constructing Lyapunov functionals $L_{\alpha,\beta}$ and proving exponential decay of the velocity, supplemented by renormalized entropy estimates to control the temperature. The analysis combines Faedo–Galerkin approximations, uniform energy and entropy bounds, weak lower semicontinuity, and careful limit passages, yielding a robust long‑time theory for a physically relevant but mathematically challenging regime. These results extend the stability theory for non‑Newtonian, heat‑conducting fluids into the supercritical regime and provide a framework for understanding energy exchange with boundaries through a corrected energy/entropy structure.

Abstract

We consider a three-dimensional domain occupied by a homogeneous, incompressible, non-Newtonian, heat-conducting fluid with prescribed nonuniform temperature on the boundary and no-slip boundary conditions for the velocity. No external body forces are assumed. The constitutive relation for the Cauchy stress tensor is assumed in a general form that includes, in particular, the power-law and Ladyzhenskaya models with the power-law exponent in the range where neither regularity, uniqueness, nor the validity of the energy equality is known to hold. Nevertheless, we introduce a novel concept of solution suitable for this setting, which enables us to establish the existence of global-in-time solutions for arbitrary physically relevant initial data. A remarkable feature of this formulation is that the steady-state solution is nonlinearly stable: every such solution converges, in a suitable sense, to the steady state as time tends to infinity. This provides the first result that combines existence with long-time stability in this physically relevant yet mathematically challenging regime.

On the stability of solutions to non-Newtonian Navier--Stokes--Fourier-like systems in the supercritical case

TL;DR

This work analyzes a three‑dimensional, incompressible non‑Newtonian, heat‑conducting fluid in a Lipschitz domain, modeled by a Navier–Stokes–Fourier–type system with nonuniform boundary temperature and a general ‑growth constitutive law in the supercritical range. To handle potential lack of regularity and energy equality, the authors introduce a novel ‑weak solution framework built around a ‑correction function and a ‑corrected total energy inequality, together with an entropy inequality, and prove global existence for and . They establish nonlinear stability of the unique steady state by constructing Lyapunov functionals and proving exponential decay of the velocity, supplemented by renormalized entropy estimates to control the temperature. The analysis combines Faedo–Galerkin approximations, uniform energy and entropy bounds, weak lower semicontinuity, and careful limit passages, yielding a robust long‑time theory for a physically relevant but mathematically challenging regime. These results extend the stability theory for non‑Newtonian, heat‑conducting fluids into the supercritical regime and provide a framework for understanding energy exchange with boundaries through a corrected energy/entropy structure.

Abstract

We consider a three-dimensional domain occupied by a homogeneous, incompressible, non-Newtonian, heat-conducting fluid with prescribed nonuniform temperature on the boundary and no-slip boundary conditions for the velocity. No external body forces are assumed. The constitutive relation for the Cauchy stress tensor is assumed in a general form that includes, in particular, the power-law and Ladyzhenskaya models with the power-law exponent in the range where neither regularity, uniqueness, nor the validity of the energy equality is known to hold. Nevertheless, we introduce a novel concept of solution suitable for this setting, which enables us to establish the existence of global-in-time solutions for arbitrary physically relevant initial data. A remarkable feature of this formulation is that the steady-state solution is nonlinearly stable: every such solution converges, in a suitable sense, to the steady state as time tends to infinity. This provides the first result that combines existence with long-time stability in this physically relevant yet mathematically challenging regime.
Paper Structure (27 sections, 4 theorems, 216 equations)

This paper contains 27 sections, 4 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Functional setting
  3. Definition of the notion of solution
  4. Main results
  5. Existence of b - the proof of Theorem \ref{['thm:ex']}
  6. Faedo--Galerkin approximations
  7. Inequalities for approximations
  8. Kinetic energy inequality
  9. Total energy balance
  10. Entropy balance
  11. Limit in the Faedo--Galerkin approximations
  12. A priori $n$-independent estimates
  13. Uniform $T$-independent estimates for $\vartheta$
  14. Weak lower semicontinuity of $\mathbf{S} : \mathbf{D}{\bf v}$ and the limit in the kinetic energy inequality
  15. Limit in the $b$ - corrected total energy inequality
  16. ...and 12 more sections

Key Result

Theorem 1.3

Let $\gamma\in(-1,-1/2)$ and let $b$ be a fixed $\gamma$-correction function. Assume that the data $\hat{\vartheta}$ satisfy MB6, and the initial data ${\bf v}_0$ and $\vartheta_0$ satisfy MB7 and MB7.5, respectively. Assume further that $\kappa$ and $\mathbf{S}^*$ satisfy MBBounds--s:monotone with

Theorems & Definitions (7)

  • Definition 1.1: $\gamma$-correction function
  • Definition 1.2: $b$-weak solution
  • Theorem 1.3
  • Theorem 1.4
  • Lemma A.1: Decay estimates
  • proof
  • Theorem A.2